| 11. | Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide.
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| 12. | For example, every quasi-projective scheme over a Noetherian ring has the resolution property.
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| 13. | For commutative Noetherian rings, this is the same as the definition using chains of prime ideals.
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| 14. | That follows because the rings of algebraic geometry, in the classical sense, are Noetherian rings.
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| 15. | Both facts imply that a finitely generated commutative algebra over a Noetherian ring is again a Noetherian ring.
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| 16. | Both facts imply that a finitely generated commutative algebra over a Noetherian ring is again a Noetherian ring.
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| 17. | This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension.
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| 18. | All quotient rings of a Noetherian ring are Noetherian, but that does not necessarily hold for its subrings.
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| 19. | Over a commutative Noetherian ring, this gives a particularly nice understanding of all injective modules, described in.
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| 20. | A "'Gorenstein ring "'is a commutative Noetherian ring such that each Cohen Macaulay.
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